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X C Now let c be a real number such that c 0 Then f c c Also lim f x lim x c X C X C Since lim f x f c f is continuous is continuous at all points Example 8 Discuss the continuity of the function f given by f x x x 1 Solution Clearly fis defined at every real number c and its value at c is c c 1 We also know that fis a continuous function Thus lim f x f c and hence fis continuous at every real number This means X C Solution Fix any non zero real number c we have 1 1 Example 9 Discuss the continuity of the function f defined by f x x 0 X lim x x 1 c c 1 X C explanatory lim f x lim X C X f x X C X 1 Also since for c 0 f c we have lim f x f c and hence fis continuous X C at every point in the domain off Thus fis a continuous function 1 1 3 333 Rationalised 2023 24 0 3 0 2 5 We take this opportunity to explain the concept of infinity This we do by analysing 1 X the function f x near x 0 To carry out this analysis we follow the usual trick of finding the value of the function at real numbers close to 0 Essentially we are trying to find the right hand limit of fat 0 We tabulate this in the following Table 5 1 Table 5 1 0 3 0 2 0 1 10 0 01 10 5 10 100 10 0 001 10 10 f x 13 333 1000 10 10 We observe that as x gets closer to 0 from the right the value of f x shoots up higher This may be rephrased as the value of f x may be made larger than any given number by choosing a positive real number very close to 0 In symbols 1 lim f x x 0 to be read as the right hand limit of f x at 0 is plus infinity We wish to emphasise that is NOT a real number and hence the right hand limit of fat 0 does not exist as a real number Similarly the left hand limit of f The following table is self hed C real numbers Hence f CONTINUITY AND DIFFERENTIABILITY 109 Table 5 2 0 may be fou 10 10 10 10 3 10 10 2 10

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